A note on Büchi's problem for p-adic numbers
DOI:
https://doi.org/10.4067/S0716-09172011000300002Abstract
We prove that for any prime p and any integer k > 2,there exist in the ring Zp of p-adic integers arbitrarily long sequences whose sequence of k-th powers 1) has its k-th difference sequence equal to the constant sequence (k!); and 2) is not a sequence of consecutive k-th powers. This shows that the analogue of Buchi's problem for higher powers has a negative answer over Zp.This result for k = 2 was recently obtained by J. Browkin.
Downloads
References
[2] H. Hasse, Number Theory, Springer (1980).
[3] N. Koblitz, P-adic Numbers, p-adic Analysis, and Zeta-Functions, Springer Graduate Texts in Mathematics, (1996).
[4] J. Neukirch, Class field theory, Springer Verlag, Grundlehren der mathematischen Wissenschaften 280, A Series of Comprehensive Studies in Mathematics, (1986).
[5] H. Pasten, T. Pheidas and X. Vidaux, A survey on Büchi’s problem: new presentations and open problems, Zapiski Nauchn. Sem. POMI 377, pp. 111-140, (2010).
[6] T. Pheidas and X. Vidaux, Extensions of Büchi’s problem : Questions of decidability for addition and n-th powers, Fundamenta Mathematicae 185, pp. 171-194, (2005).
Downloads
Published
Issue
Section
License
-
Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.